The text I am using "Nonlinear PDEs - A Dynamical Systems Approach" (Hannes Uecker) defines a stable manifold as follows
Definition : Let $u^*$ be a fixed point of the ODE $\dot{u} = f(u)$ with solutions denoted $u(t,u_0)$ with $u(t_0) = u_0$. The set $$W_s(u^*) := \{ u_s\in \mathbb{R}^d \ | \ \exists \beta>0, \lim_{t\rightarrow \infty} ||u(t,u_s) - u^*||_{\mathbb{R}^d}e^{\beta t} =0 \}$$ is called the stable manifold
Then this follows with the stable manifold theorem which states that a unique manifold which is tangent to the stable eigenspace exists and agrees with the above definition. However I am wondering about the following phase diagram
It appears that the stable manifold should be $\mathbb{R}^2 - (\mathbb{R}\times \{0\})$ because each point in this set is attracted to the the origin as $t\rightarrow \infty$ but only the space $W_s(u^*) = E^s = \{0\}\times\mathbb{R}$ is tangent to $E^s$. Should the theorem state that a subset of $W_s(u^*)$ will be tangent to $E^s$ or am I missing something here? I feel like I am missing something in the definition as the the curved lines become tangent to the center eigenspace $E^c$ (which does not have a unique tangent center manifold).